A Fourier (k-) space design approach for controllable photonic band and localization states in aperiodic lattices

In this paper we present a systematic study of photonic bandgap engineering using aperiodic lattices (ALs). Up to now ALs have tended to be defined by specific formulae (e.g. Fibonacci, Cantor), and theories have neglected other useful ALs along with the vast majority of non-useful (random) ALs. Here, we present a practical and efficient Fourier space-based general theory to identify all those ALs having useful band properties, which are characterized by well-defined Fourier (i.e. lattice momentum) components. Direct control of field localization comes via control of the Parseval strength competition between the different Fourier components characterizing a lattice. Real-space optimization of ALs tends to be computationally demanding. However, via our Fourier space-based simulated annealing inverse optimization algorithm, we efficiently tailor the relative strength of the AL Fourier components for precise control of photonic band and localization properties.